PRESERVATION THEOREMS FOR AE-SENTENCES
Diego Castaño, Miguel A. Campercholi, Diego Vaggione · Journal of Symbolic Logic · 2025
Abstract An AE-sentence is a sentence of the form $\forall x_1\ldots x_n \exists ! z_1\ldots z_m \, \alpha (\mathbf {x},\mathbf {z})$ , where $\alpha $ is a finite conjunction of equations. Given an arbitrary quasivariety $\mathcal {Q}$ , we give a purely semantical characterization of when a class $\mathcal {K} \subseteq \mathcal {Q}$ with $S(\mathcal {K}) = \mathcal {Q}$ is axiomatizable relative to $\mathcal {Q}$ by AE-sentences. Along the way, we also characterize axiomatizability by generalized AE-sentences , which are of the form described above, except that both the number of existential quantifiers and of equations in $\alpha $ are allowed to be infinite. The article concludes with an analysis of how the main theorems can be improved in the context of finitely generated quasivarieties.